Bonn Summer SchoolAdvances in Empirical Macroeconomics

Karel Mertens

Cornell, NBER, CEPR

Bonn, June 2015

In God we trust, all others bring data.

William E. Deming (1900-1993)

Angrist and Pischke are Mad About Macro

(2010 JEP, The Credibility Revolution in Empirical Economics: HowBetter Research Design Is Taking the Con out of Econometrics)

Sims, JEP 2010, Comment on Angrist and Pischke: But economics is notan experimental science

Overview

1. Estimating the Effects of Shocks Without Much Theory

1.1 Structural Time Series Models1.2 Identification Strategies

2. Applications to Fiscal Shocks

2.1 Tax Policy Shocks2.2 Government Spending Shocks2.3 Austerity Measures

3. Two Difficulties in Interpreting SVARs

3.1 Noninvertibility3.2 Time Aggregation

4. Systematic Tax Policy and the ZLB

1. Estimating the Effects of Shocks Without Much Theory

General Approach:

1. Using observables zt , fit a model of expectations E [zt | It−1]

2. Identify meaningful shocks from innovations zt − E [zt | It−1].

3. Estimate dynamic causal effects of shocks/variance contributions.

It : information available to economic decision makers at time t.

1.1 Structural Time Series Models

State Space (SS) Representation

The solution of stationary linear models can generally be written as

st = Gst−1 + Fetzt = Ast−1 +Det

st is a m × 1 vector of state variableset+1 is an l × 1 vector of uncorrelated white noise, or structural shocks

E [et ] = 0 , E [ete′t ] = I , E [ete

′s ] = 0 for s 6= t

zt is an n × 1 vector of variables of interestG is m ×m, F is m × l , A is n ×m and D is n × l

Stability and Stationarity

The m ×m matrix G has all eigenvalues less than one in modulus, i.e.

det(M − λI ) 6= 0 for | λ |≥ 1, or equivalently

det(I −Mz) 6= 0 for | z |≤ 1

st follows a stable VAR(1) process

st and zt are stationary stochastic processes, i.e. the first and secondmoments are time invariant.

Lag Operator

Define the lag operator L, i.e. Lkxt = xt−k .

st = Gst−1 + Fet(I − GL)st = Fet

st = (I − GL)−1Fet

st =∞∑i=0

G iFet−i

Time Series Representations

Moving Average (MA) Representation

MA(q) : zt = M(L)υt =

q∑i=0

Miυt−i

where M(L) =M0 +M1L + ...+MqLq and innovations process υt

E [υt ] = 0 , E [υtυ′t ] = Σ , E [υtυ

′s ] = 0 for s 6= t

Wold Representation TheoremEvery stationary process zt can be written as an MA(∞).(plus deterministic terms).

Stationary linear models for zt and et can always be written as MA(∞)

zt = M∗(L)et =∞∑i=0

M∗i et−i

Given an SS representation G,F ,A,D,

zt =∞∑i=1

AG i−1Fet−i +Det

= A(I − GL)−1Fet−1 +Det=

(D +A(I − GL)−1FL

)et

such that M∗0 = D and M∗i = AG i−1F for i ≥ 1

Assume n = l and Stochastic Nonsingularity:

D is an n × n invertible matrix.

We can write a ‘structural’ MA

zt = M(L)υt =∞∑i=0

Miυt−i

with υt = Det , Σ = DD′ , M0 = I

zt =(I +A(I − GL)−1FD−1L

)υt

such that Mi = AG i−1FD−1 for i ≥ 1

Time Series Representations

Vector Autoregressive Moving Average Representation

VARMA(p,q) : B(L)zt = M(L)υt

where

B(L) = I − B1L− ...− BpLp

M(L) = M0 +M1L + ...+MqLq

E [υt ] = 0 , E [υtυ′t ] = Σ , E [υtυ

′s ] = 0 for s 6= t

Starting from the MA representation of our linear models,

zt = A(I − GL)−1Fet−1 +Det

Suppose n = m and A is invertible,

A−1zt = (I − GL)−1Fet−1 +A−1Det(I − GL)A−1zt = Fet−1 + (I − GL)A−1Det

Assuming D invertible we obtain a structural VARMA(1,1),

zt = AGA−1zt−1 + υt −A(G − FD−1A

)A−1υt−1

’Structural’ here means

υt = Det

Time Series Representations

Vector Autoregressive Representation

VAR(p) : B(L)zt = υt

⇒ zt = B1zt−1 + . . .+ Bpzt−p + υt

where

B(L) = I − B1L− ...− BpLp

E [υt ] = 0 , E [υtυ′t ] = Σ , E [υtυ

′s ] = 0 for s 6= t

Fernandez-Villaverde, Rubio-Ramirez, Sargent and Watson (2007):

Start from the SS representation of our models,

st = Gst−1 + Fetzt = Ast−1 +Det

Assume n = l and Stochastic Nonsingularity:

D is an n × n invertible matrix.

When D is nonsingular,

et = D−1(zt −Ast−1)

Substituting

st =(G − FD−1A

)st−1 + FD−1zt

Invertibility (in the Past):

The eigenvalues of G − FD−1A are strictly less then one in modulus.

Under this condition we can write

st =∞∑i=0

(G − FD−1A

)i FD−1zt−i

Substituting

zt =∞∑i=1

A(G − FD−1A

)i−1 FD−1zt−i +Det

such that

Bi = A(G − FD−1A

)i−1 FD−1υt = Det

In practice, lag truncation: zt =∑p

i=1 Bizt−i + υt

Stochastic Nonsingularity: No big deal (measurement errors)

Invertibility (in the Past): Choice of zt is important!If the invertibility condition does not hold and we estimate a VAR:

υt 6= Det

Alternatively, start from the MA representation

zt = M(L)υt

or VARMA representation

B(L)zt = M(L)υt

and invert M(L) to obtain a VAR representation.

(Note: it is assumed that M(L) is a square matrix of rational functions.)

S(L)B(L)zt = S(L)M(L)υt

M(L) is invertible in the past,

i.e. there is an S(L) such that S(L)M(L) = I and S(L) only hasnonnegative powers of L,

if det(M(L)) 6= 0 for | L |≤ 1.

See Hansen and Sargent (1980, 1991), Lippi and Reichlin (1993), Forniand Gambetti (2014)

MA representation of our models:

zt =(I +A(I − GL)−1FD−1L

)υt

Using the matrix determinant lemma,

det(I +A(I − GL)−1FD−1L

)= det

(I − (G − FD−1A)L

)/det(I − GL)

The condition that

det(I − (G − FD−1A)L

)6= 0 for | L |≤ 1

is equivalent to

det((G − FD−1A)− Iz

)6= 0 for | z |≥ 1

or G − FD−1A must have all eigenvalues strictly less then one inmodulus (same as Fernandez-Villaverde et al. 2007).

VARMA representation of our models:

(I −AGA−1L)zt =(I −A

(G − FD−1A

)A−1L

)υt−1

The condition that

det(I −A

(G − FD−1A

)A−1L

)6= 0 for | L |≤ 1

is equivalent to

det((G − FD−1A

)− Iz

)6= 0 for | z |≥ 1

or G − FD−1A must have all eigenvalues strictly less then one inmodulus (same as Fernandez-Villaverde et al. 2007).

Example: New Keynesian Model

See Clarida, Gali and Gertler (1999) and the Woodford (2003) and Gali(2008) books

Et∆ygapt+1 = φππt − Etπt+1 − ut (Eq. Euler)

πt = κygapt + βEtπt+1 − vt (Phillips curve)

where κ > 0, φπ > 1, 0 ≤ β < 1

ygapt : output gap , πt : inflation

vt : cost push shocks

ut : other shocks (technology, govt spending, taxes, monetary policy,...)

ut and vt are stationary exogenous processes

Iterating forward,[ygapt

πt

]= Et

∞∑j=0

C−(j+1)

[ut+j

vt+j

]where

C−1 =1

1 + φπκ

[1 1− βφπκ β + κ

]Note C−1 has eigenvalues strictly less than one in modulus. Why?

Suppose the shocks follow a VAR(1) process.[utvt

]︸ ︷︷ ︸

st

= Λ︸︷︷︸G

[ut−1vt−1

]+ Ω︸︷︷︸F

et

[ygapt

πt

]︸ ︷︷ ︸

zt

=∞∑j=0

C−(j+1)Λj

[utvt

]

= (C − Λ)−1 Λ︸ ︷︷ ︸A

[ut−1vt−1

]︸ ︷︷ ︸

st−1

+ (C − Λ)−1 Ω︸ ︷︷ ︸D

et

Note that G − FD−1A = 0.

State Space representation:[utvt

]= Λ

[ut−1vt−1

]+ Ωet[

ygapt

πt

]= (C − Λ)−1 Λ

[ut−1vt−1

]+ (C − Λ)−1 Ωet

Moving Average Representation[ygapt

πt

]=∞∑i=0

(C − Λ)−1 ΛiΩet−i

VAR/VARMA Representation[ygapt

πt

]= (C − Λ)−1 Λ (C − Λ)

[ygapt−1πt−1

]+ (C − Λ)−1 Ωet

Reduced Form Parameters

SS : st = Gst−1 +Hυt G,A,H,Σzt = Ast−1 + υt

MA(q) : zt = M(L)υt Mi ,Σ

VARMA(p,q) : B(L)zt = M(L)υt Bi ,Mi ,Σ

VAR(p) : B(L)zt = υt Bi ,Σ

where E [υt ] = 0 , E [υtυ′t ] = Σ , E [υtυ

′s ] = 0 for s 6= t

Note: SS, MA and VARMA require additional normalizations.

When well-specified, these are all models that allow us to separateexpectations E [zt | It−1] and innovations υt = zt − E [zt | It−1].

Estimation of Reduced Form Parameters

VAR estimation

Some references:

Hamilton, 1994, ’Time Series Analysis’

Luetkepohl, 2005, ‘A New Introduction to Time Series Analysis’

Brockwell and Davis, 2006,‘Time Series: Theory and Methods’

Aoki, 1990, ‘State Space Modeling of Time Series’

Durbin and Koopman, 2012, ‘Time Series Analysis by State SpaceMethods’

Local Uniquess

In matrix form

Et

[ygapt+1

πt+1

]= C

[ygapt

πt

]−[

utvt

], C ≡ 1

β

[β + κ βφπ − 1−κ 1

]The companion matrix C has the characteristic polynomial

P(ϕ) = ϕ2 − tr(C)ϕ+ det(C)

tr(C) = 1 + 1/β + κ/β > 1

det(C) = (1 + κφπ)/β > 1

which has roots outside the unit circle if tr(C) < 1 + det(C) or

φπ > 1 (Taylor Principle)

Back

Estimating a VARSample of T + p observations of an n × 1 vector zt :

zt−p+1, zt−p+2, . . . , zT−1, zT

Define the n × T matrix z such that:

z =[z1 z2 · · · zT

]Define a np × 1 vector Zt :

Zt =

zt−1zt−2

...zt−p

Let Z be a np × T matrix collecting T observations of Zt :

Z =[Z1 Z2 · · · Zp

]

Let υ be a n × T matrix of n × 1 residuals υt :

υ =[υ1 υ2 · · · υT

]Let B be a n × np matrix of coefficients:

B =[B1 B2 · · · Bp

]Introduce the vectorization operator:

z = vec(z)

u = vec(υ)

Where z is a nT × 1 vector of the stacked columns of z . Thevariance-covariance matrix of u is:

Var(υ) ≡ Σ = IT ⊗ Σ

Generalized Least Squares

Re-write the reduced form VAR(p) as:

z = BZ + u

Or as:

z = (Z ′ ⊗ In)β + u

Where ⊗ is the Kronecker product. We can estimate β with GeneralizedLeast Squares (GLS):

u′Σ−1u = (z− (Z ′ ⊗ In)β)′Σ−1(z− (Z ′ ⊗ In)β)

= z′Σ−1z + β′(Z ⊗ In)Σ−1(Z ′ ⊗ In)β − 2β′(Z ⊗ In)Σ−1z

= z′(IT ⊗ Σ−1)z + β′(ZZ ′ ⊗ Σ−1)β − 2β′(Z ⊗ Σ−1)z

First order condition:

2(ZZ ′ ⊗ Σ−1)β − 2(Z ⊗ Σ−1)z = 0

The GLS estimator is therefore:

β = ((ZZ ′)−1Z ⊗ In)z

This is the same as OLS or ML.

Asymptotic normality:

√T (β − β)

d→ N (0, Γ−1 ⊗ Σ)

where Γ = plimZZ ′/T and the estimators are

Γ =ZZ ′

T

Σ =1

T − np − 1z(IT − Z ′(ZZ ′)−1Z )z ′

Back